Instanton Homology and Symplectic Fixed Points

Instanton homology and symplectic ﬁxed points

Stamatis Dostoglou and Dietmar Salamon∗ Mathematics Institute University of Warwick Coventry CV4 7AL Great Britain September 1992

1 Introduction

A gradient flow of a Morse function on a compact Riemannian manifold is said to be of Morse-Smale type if the stable and unstable manifolds of any two critical points intersect transversally. For such a Morse-Smale gradient flow there is a chain complex generated by the critical points and graded by the Morse index. The boundary operator has as its (x, y)-entry the number of gradient flow lines running from x to y counted with appropriate signs whenever the difference of the Morse indices is 1. The homology of this chain complex agrees with the homology of the underlying manifold M and this can be used to prove the Morse inequalities [30] (see also [24]). Around 1986 Floer generalized this idea and discovered a powerful new approach to infinite dimensional Morse theory now called Floer homology. He used this approach to prove the Arnold conjecture for monotone symplectic manifolds [12] and discovered a new invariant for homology 3-spheres called instanton homology [11]. This invariant can roughly be described as the homology of a chain complex generated by the irreducible representations of the fundamental group of the homology 3-sphere M in the Lie group SU(2). These representations can be thought of as flat connections on the principal bundle M × SU(2) and they appear as the critical points of the Chern-Simons functional on the infinite dimensional configuration space of connections on this bundle modulo gauge equivalence. The gradient flow lines of the Chern-Simons functional are the self-dual Yang-Mills instantons on the 4-manifold M × R and

∗This research has been partially supported by the SERC.

1 they determine the boundary operator of instanton homology. The instanton inst homology groups are denoted by HF∗ (M). They play an important role in Donaldson’s theory of 4-manifolds. If X is a 4-manifold whose boundary M = ∂X is a homology-3-sphere then the Donaldson-polynomials of X take their values in the instanton homology groups of M. Via a Heegard splitting a homology-3-sphere can be represented as the union

M = M0 ∪Σ M1 of two handle bodies whose common boundary Σ = ∂M0 = ∂M1 is a Riemann surface. The moduli space of irreducible flat connections on Σ × SU(2) is a finite dimensional symplectic manifold M (with singularities) and the subset of those which extend to a flat connection on Mi form a Lagrangian submanifold Li ⊂ M for i = 0, 1. The intersection points of these Lagrangian submanifolds correspond precisely to the flat connections on M that is the critical points of the Chern-Simons functional. Now there is a Floer theory for Lagrangian intersections in which the critical points are the intersection points of the La- grangian submanifolds and the connecting orbits are pseudo-holomorphic curves u : [0, 1] × R → M which satisfy u(i, t) ∈ Li and converge to intersection points x ∈ L0 ∩ L1 as t tends to ∞. The associated Floer homology groups are denoted by symp HF∗ (M, L0, L1). In [1] Atiyah conjectured that the instanton homology of a homology-3-sphere M is isomorphic to the Floer homology of the triple M, L0, L1 associated to a Heegard splitting as above

inst symp HF∗ (M) = HF∗ (M, L0, L1). In this paper we address a similar but somewhat simpler problem which was suggested to us by A. Floer. The instanton homology groups can also be constructed for 3-manifolds which are not homology-3-spheres if instead of the trivial SU(2)-bundle we take a nontrivial SO(3)-bundle Q over M. Then the integral first homology of M is necessarily nontrivial and we assume that there is no 2-torsion in H1. This assumption guarantees that every flat connection on Q has a discrete isotropy subgroup and is therefore a regular point for the action of the identity component of the group of gauge transformations on the space of connections. The instanton homology groups of the bundle Q are denoted by inst HF∗ (M; Q). A special case is where the bundle Q = Pf is the mapping cylinder of a nontrivial SO(3)-bundle π : P → Σ over a Riemann surface Σ for an automorphism f : P → P . The underlying 3-manifold is the mapping cylinder M = Σh of Σ for the diffeomorphism h : Σ → Σ induced by f. Then the flat connections on Pf correspond naturally to the fixed points of the symplectomorphism

φf : M(P ) → M(P )

2 induced by f on the moduli space M(P ) of flat connections on the bundle P . This moduli space is a compact symplectic manifold (without singularities) of dimension 6k − 6 where k ≥ 2 is the genus of Σ. It is well known that this manifold is connected and simply connected [21] and in [2] Atiyah and Bott proved that π2(MF (P )) = Z. For any symplectomorphism φ : M → M of such symp a symplectic manifold there are Floer homology groups HF∗ (M, φ). In this theory the critical points are the fixed points of φ and the conecting orbits are pseudoholomorphic curves u : R2 → M which satisfy u(s+1, t) = φ(u(s, t)) and converge to fixed points x of φ as t tends to ∞. The Euler characteristic of symp HF∗ (M, φ) is the Lefschetz number of φ. The main result of this paper is the following. Theorem 1.1 There is a natural isomorphism of Floer homologies

symp inst HF∗ (M(P ), φf ) = HF∗ (Σh; Pf ). The proof consists of two steps. The ﬁrst is an index theorem about the spectral ﬂows of two associated families of self adjoint operators and states that the relative Morse indices agree. The second step is a characterization of holomorphic curves in the moduli space M(P ) as a limit case of self-dual Yang-Mills instantons on the bundle Pf × R over the 4-manifold Σh × R. In the present paper we outline the main ideas of the proof. Details of the analysis will appear elsewhere. Thanks to P. Braam, S. Donaldson, J.D.S. Jones, and J. Robbin for helpful discussions.

2 Instanton homology

In this section we discuss Floer’s instanton homology for nontrivial SO(3)- bundle. All the theorems are due to Floer [11], [13]. Let M be a compact connected oriented 3-dimensional manifold without boundary and π : Q → M be a principal bundle with structure group G = SO(3). We identify the space A(Q) of connections on Q with the space of smooth Lie algebra valued 1-forms a ∈ Ω1(Q, g) which are equivariant with respect to the adjoint action of G and canonical in the ﬁbres:

−1 apx(vx) = x ap(v)x, ap(pξ) = ξ for v ∈ TpQ, x ∈ G, and ξ ∈ g. The space G(Q) of gauge transformations can be canonically identiﬁed with the space smooth maps g : Q → G which are equivariant under the action of G on itself through inner automorphisms: g(px) = x−1g(p)x.

Thus gauge transformations are sections of the adjoint bundle GQ = Q ×ad G which consists of equivalence classes of pairs [p, A] where p ∈ Q and g ∈ G under

3 the equivalence relation [p, g] ≡ [px, x−1gx] for x ∈ G. With these identiﬁcations G(Q) acts on A(Q) by the formula

g∗a = g−1dg + g−1ag.

Let G0(Q) denote the component of the identity in G(Q) and consider the quotient C(Q) = A(Q)/G0(Q).

This space is not an inﬁnite dimensional manifold since the group G0(Q) does not act freely on A(Q). However, almost every connection is a regular point of the action. More precisely, let

∗ Ga = {g ∈ G(Q) : g a = a} denote the isotropy subgroup of a connection a ∈ A(Q).

Lemma 2.1 The isotropy subgroup of a connection a ∈ A(Q) is discrete if and only if Ga ∩ G0(Q) = {1}. Connections with a discrete isotropy subgroup are called regular.

Proof: Suppose that g ∈ Ga ∩ G0(Q) and g 6= 1. Then g lifts to a map g˜ : Q → SU(2) and g˜(p) ∈ SU(2) commutes with the holonomy subgroup H˜ ⊂ SU(2) of a at the point p ∈ Q lifted to SU(2). Since g 6= 1 the matrix g˜(p) has two distinct eigenvalues and both eigenspaces are invariant under the holonomy group H˜ . Thus there exists a circle of matrices in SU(2) commuting with H˜ and therefore Ga is not discrete. 2 The group of components of G(Q) acts on C(Q). These components are characterized by two invariants, the degree

deg : G(Q) → Z and the parity 1 η : G(Q) → H (M; Z2). The deﬁnition of the degree is based on the next lemma. The proof was per- sonally communicated to the second author by John Jones.

Lemma 2.2 (1) If w2(Q) = 0 then the induced map π∗ : H3(Q, Z) → H3(M, Z) is onto.

∗ 3 3 (2) If w2(Q) 6= 0 then the induced map π : H (M, Z2) → H (Q, Z2) is zero and H3(M, Z)/im π∗ = Z2.

(3) For every g ∈ G(Q) the induced map g∗ : H3(Q, Z) → H3(SO(3), Z) descends to a homomorphism H3(M, Z) → H3(SO(3), Z).

4 Proof: Examine the spectral sequence of the bundle Q → M with integer coeﬃcients to obtain

H3(Q, Z)/ ker π∗ ' im π∗ ' ker d2 : H3(M, Z) → H1(M, Z2) .

Now the image of the fundamental class [M] under d2 : H3(M, Z) → H1(M, Z2) is the Poincar´e dual of w2(Q). Hence π∗ is onto if and only if w2(Q) = 0 and im π∗ = h2[M]i otherwise. This proves statements (1) and (2). Now let ι : SO(3) → Q be the inclusion of a ﬁber and denote the induced map on homology by ι∗ : H3(SO(3), Z) → H3(Q, Z). Then it follows again by examining the spectral sequence that ι∗ is injective and

im ι∗ ⊂ ker π∗, ker π∗/im ι∗ ' H2(M, Z2)

For every gauge transformation g : Q → SO(3) denote g∗ : H3(Q, Z) → H3(SO(3), Z). The composition g ◦ ι is homotopic to a constant and hence im ι∗ ⊂ ker g∗. This implies that g∗ descends to a homomorphism H3(Q, Z)/im ι∗ → H3(SO(3), Z) = Z. Any such homomorphism must vanish on the subgroup ker π∗/im ι∗ ' H2(M, Z2). Thus we have proved that

ker π∗ ⊂ ker g∗ for every gauge transformation g ∈ G(Q). In view of statement (2) it suﬃces to prove that w2(Q) 6= 0 =⇒ im g∗ ⊂ h2[SO(3)]i for every gauge transformation g ∈ G(Q). We prove instead that the dual ∗ 3 3 homomorphism g : H (SO(3), Z2) → H (Q, Z2) is zero. To see this consider 1 ∗ ∗ ∗ ∗ the generator α ∈ H (SO(3), Z2). Since ι g = 0 it follows that g α = π β 1 ∗ 3 ∗ 3 for some β ∈ H (M, Z2). Hence g (α ) = π (β ) = 0 where the last assertion follows from statement (2). This proves the lemma. 2

For g ∈ G(Q) the induced homomorphism H3(M, Z) = Z → H3(G, Z) = Z is determined by an integer deg(g) called the degree of g. Alternatively, the degree can be deﬁned as the intersection number of the submanifolds graph (g) = {[p, g(p)] : p ∈ Q} and graph (1l) of the adjoint bundle GQ. The homomorphism g∗ : π1(Q) → π1(G) = Z2 descends to a homomorphism η(g) : π1(M) → Z2, called the parity of g. It is the obstruction for g to lift to a map g˜ : Q → SU(2). A gauge transformation is called even if η(g) = 0. Every even gauge transformation is of even degree but not vice versa. Moreover, the 1 map η : G(Q) → H (M, Z2) is always onto. Throughout we shall assume the following hypothesis. 1 Hypothesis (H1) Every cohomology class η ∈ H (M; Z2) can be represented by ﬁnitely many embedded oriented Riemann surfaces. Moreover w2(Q) 6= 0.

5 If M is orientable then every one dimensional integral cohomology class can be represented by finitely many embedded oriented Riemann surfaces. So the first part of hypothesis (H1) will be satisfied whenever M is orientable and there is no 2-torsion in H1. Also note that (H1) implies the following weaker hypothesis. Hypothesis (H2) There exists an embedding ι : Σ → M of a Riemann surface such that ι∗Q is the nontrivial SO(3)-bundle over Σ. Some important consequences of hypotheses (H1) and (H2) are summarized in the next lemma. The proof will be given in the appendix.

Lemma 2.3 (1) Two gauge transformations are homotopic if and only if they have the same degree and the same parity. (2) Assume (H1). Then for every g ∈ G(Q)

deg(g) ≡ w2(Q) · η(g) (mod 2). (1)

1 Conversely, for every integer k and every η ∈ H (M; Z2) with k ≡ w2(Q)· η (mod 2) there exists a gauge transformation with deg(g) = k and η(g) = η. (3) Assume (H2). Then there exists a gauge transformation g of degree 1.

Remark 2.4 (i) If Q = RP 3 ×SO(3) then there exists a gauge transformation of degree 1. However, for such a gauge transformation equation (1) is violated since w2(Q) = 0. If M is a homology 3-sphere and Q = M ×SO(3) is the product bundle then equation (1) is trivially satisﬁed but there is no gauge transformation of degree 1. (ii) Another interesting example is the (unique) nontrivial SO(3)-bundle π : Q → RP 3. This bundle does not satisfy hypothesis (H2). It can be 3 represented in the form Q = S ×Z2 SO(3) with (x, A) ∼ (−x, RA) where R ∈ SO(3) is a reﬂection, i.e. R2 = 1l. A gauge transformation of this bundle is a smooth map g : S3 → SO(3) such that g(−x) = Rg(x)R. Any such map lifts to a smooth map g˜ : S3 → SU(2) and the degree of g in the above sense is the degree of the lift g˜ (or half the degree of g as a map between oriented 3-manifolds). The lift g˜ is necessarily of even degree. So in this case there is no gauge transformation of degree 1. Hence statements (2) and (3) of Lemma 2.3 are both violated.

The space A(Q) of connections on Q is an aﬃne space whose associated 1 vector space is Ω (gQ). Here gQ = Q×Gg is the vector bundle over M associated to Q via the adjoint representation of G on its Lie algebra. We think of an 1 inﬁnitesimal connection α ∈ Ω (gQ) as an invariant and horizontal 1-form on Q. The Lie algebra of G(Q) is the space of invariant g-valued functions on Q

6 0 and so may be identiﬁed with Ω (gQ). The inﬁnitesimal action of G(Q) is given by the covariant derivative

0 1 da : Ω (gQ) → Ω (gQ), daη = dη + [a ∧ η]. Thus the tangent space to the conﬁguration space C(Q) at a regular connection 1 a is the quotient Ω (gQ)/im da. The curvature of a connection is the 2-form 1 F = da + [a ∧ a] ∈ Ω2(M; g ) a 2 Q and determines a natural 1-form on the space of connections via the linear functional

α 7→ hFa ∧ αi ZM 1 on TaA(Q) = Ω (gQ). Here h , i denotes the invariant inner product on the Lie algebra g given by minus the Killing form or 4 times the trace. We shall denote this 1-form by F . The Bianchi identity asserts that F is closed. Since the affine space A(Q) is contractible this implies that F is the differential of a function. Integrating F along a path which starts at a fixed flat connection a0 ∈ A(Q) we obtain the Chern-Simons functional 1 1 CS(a + α) = hd α ∧ αi + h[α ∧ α] ∧ αi 0 2 a0 3 ZM 1 for α ∈ Ω (gQ). One can check directly that

dCS(a)α = hFa ∧ αi ZM 1 for a ∈ A(Q) and α ∈ Ω (gQ). Thus the flat connections on Q appear as the critical points of the Chern-Simons functional. Since the 1-form dCS = F is invariant and horizontal it follows that the difference CS(g∗a) − CS(a) is independent of the connection a and is locally independent of the gauge transformation g. So it depends only on the component of G(Q) and it turns out that CS(a) − CS(g∗a) = 8π2 deg(g). (2) Hence as a function on the quotient A(Q)/G(Q) the Chern-Simons functional takes values in S1. Note that this function is only well defined up to an additive constant which we have chosen such that CS(a0) = 0. Denote the space of flat connections by

Aﬂat(Q) = {a ∈ A(Q) : Fa = 0} .

For every ﬂat connection a ∈ Aﬂat(Q) there is a chain complex

0 dA 1 dA 2 dA 3 Ω (gQ) −→ Ω (gQ) −→ Ω (gQ) −→ Ω (gQ).

7 j with associated cohomology groups Ha(M). These cohomology groups can be identiﬁed with the spaces of harmonic forms

j ∗ Ha(M) = ker da ∩ ker da

∗ 2 j−1 j where da denotes the L -adjoint of da : Ω → Ω with respect to a Riemannian 1 metric on M. A flat connection is called nondegenerate if Ha (M) = 0. This is consistent with the Chern-Simons point of view since the Hessian of CS at a critical point a ∈ Aflat(Q) is given by ∗da and should be viewed as an operator on 1 j 3−j the quotient Ω (gQ)/imda. Here ∗ : Ω → Ω denotes the Hodge-∗-operator with respect to the Riemannian metric on M and the invariant inner product on g. So the flat connection a is nondegenerate if and only if the Hessian of CS at a is invertible. Note that a flat connection a is both regular and nondegenerate if and only if the extended Hessian

∗d d D = a a a d∗ 0 a 1 0 is nonsingular. Here Da is a selfadjoint operator on Ω (gQ) ⊕ Ω (gQ).

Lemma 2.5 If (H2) is satisﬁed then every ﬂat connection on Q is regular.

Proof: By (H2) there exists an embedding ι : Σ → M of an oriented Riemann ∗ surface such that w2(ι Q) 6= 0. Let a ∈ Aﬂat(Q) and g ∈ Ga ∩ G0(Q). Then ∗ ∗ ∗ ∗ ∗ ι a ∈ Aﬂat(ι Q) and ι g ∈ Gι∗a ∩ G0(ι Q). By Lemma 4.1 below ι g = 1l and hence g = 1l. 2

Remark 2.6 It is easy to construct regular ﬂat connections with Ga 6= {1l}. The holonomy of such a connection is conjugate to the abelian subgroup of diagonal matrices in SO(3) with diagonal entries 1

Via the holonomy the ﬂat connections on Q correspond naturally to representations ρ : π1(M) → SO(3). 2 The second Stiefel-Whitney class w2(Q) ∈ H (M; Z2) appears as the cohomology class 2 wρ ∈ H (π1(M), Z2) associated to ρ as follows. Choose any map ρ˜ : π1(M) → SU(2) which lifts ρ −1 −1 and deﬁne wρ(γ1, γ2) = ρ˜(γ1γ2)ρ˜(γ2) ρ˜(γ1) = 1. Then wρ is a cocycle:

∂wρ(γ1, γ2, γ3) = wρ(γ2, γ3)wρ(γ1γ2, γ3)wρ(γ1, γ2γ3)wρ(γ1, γ2) = 1.

The coboundaries are functions of the form

∂f(γ1, γ2) = f(γ1)f(γ1γ2)f(γ2)

8 for some map f : π1(M) → {1}. Hence wρ is a coboundary if and only if the lift ρ˜ : π1(M) → SU(2) can be chosen to be a homomorphism. The cohomology class of wρ is independent of the choice of the lift ρ˜. It determines the second Stiefel-Whitney class of the bundle Q via the natural homomorphism 2 2 ι : H (π1(M), Z2) → H (M, Z2). The Casson invariant λ(M; Q) of the Manifold M (with respect to the bundle Q) can roughly be defined as “half the number of representations ρ : π1(M) → SO(3) with ι(wρ) = w2(Q)” or “half the number of zeros of F ”. Here the flat connections are to be counted modulo even gauge equivalence and with appropriate signs. This is analogous to the Euler number of a vector field on a finite dimensional manifold. As in finite dimensions we will in general have to perturb the 1-form F to ensure finitely many nondegenerate zeros. In [11] Floer discovered a refinement of the Casson invariant which is called instanton homology. These homology groups result from Floer’s new approach to infinite dimensional Morse theory applied to the Chern-Simons functional. The Casson invariant appears as the Euler characteristic of instanton homology. The L2-gradient of the Chern-Simons functional with respect to the Rieman- nian metric on M and the invariant inner product on g is given by grad CS(a) = 1 ∗Fa ∈ Ω (gQ). Thus a gradient flow line of the Chern-Simons functional is a smooth 1-parameter family of connections a(t) ∈ A(Q) satisfying the nonlinear partial differential equation a˙ + ∗Fa = 0. (3) The path a(t) of connections on Q can also be viewed as a connection on the bundle Q × R over the 4-manifold M × R. In this interpretation (3) is precisely the self-duality equation with respect to the product metric on M ×R. Moreover, the Yang-Mills functional agrees with the flow energy ∞ 1 2 2 YM(a) = 2 ka˙ k + kFak dt. −∞ Z The key obstacle for Morse theory in this context is that equation (3) does not define a well posed initial value problem and the Morse index of every critical point is infinite. In [11] Floer overcame this difficulty by studying only the space of bounded solutions of (3) and constructing a chain complex as was done by Witten [30] in finite dimensional Morse theory (see also [24]). In order to describe how this works we make another assumption on the bundle Q. Hypothesis (H3) Every flat connection on Q is nondegenerate. This condition will in general not be satisfied. If there are degenerate flat connections then we perturb the Chern-Simons functional in order to ensure nondegenerate critical points for the perturbed functional. If (H3) is satisfied then for every smooth solution a(t) ∈ A(Q) of (3) with finite Yang-Mills action there exist flat connections a ∈ Aflat(Q) such that a(t) converges exponentially with all derivatives to a as t tends to ∞ (see

9 for example [18]). Conversely, if a(t) is a solution of (3) for which these limits exist then the Yang-Mills action is ﬁnite:

YM(a) = CS(a−) − CS(a+).

Fix two ﬂat connections a ∈ Aﬂat(Q) and consider the space of those solutions a(t) ∈ A(Q) of (3) which also satisfy

∗ lim a(t) = ga , g ∈ G0(Q). (4) t→∞

These solutions are usually termed instantons or connecting orbits from a− to a+. The moduli space of these instantons is denoted by

{a(t) ∈ A(Q) : (3), (4)} M(a−, a+) = . G0(Q)

The next theorem due to Floer [11] summarizes some key properties of these moduli spaces.

Theorem 2.7 Assume (H2) and (H3). For a generic metric on M the moduli space M(a−, a+) is a finite dimensional oriented paracompact manifold for every pair of flat connections a ∈ Aflat(Q). There exists a function µ : Aflat(Q) → Z such that dim M(a−, a+) = µ(a−) − µ(a+). This function µ satisfies

µ(a) − µ(g∗a) = 4 deg(g) for a ∈ Aﬂat(Q) and g ∈ G(Q).

In our context the integer µ(a) plays the same role as the Morse index does in finite dimensional Morse theory. The number µ(a−) − µ(a+) is given by the spectral flow [3] of the operator family Da(t) as the connection a(t) ∈ A(Q) − + runs from a to a . So the function µ : Aflat(Q) → Z is only defined up to an additive constant. We will choose this constant such that µ(a0) = 0 where a0 ∈ Aflat(Q) is a fixed flat connection. To construct the instanton homology groups we can now proceed as in finite dimensional Morse theory [30]. The key idea is to construct a chain complex over the flat connections and to use the instantons to construct a boundary operator. For simplicity of the exposition we restrict ourselves to coefficients in Z2. Let C be the vector space over Z2 generated by the flat connections modulo gauge equivalence. For now we divide only by the component of the identity

10 G0(Q). This vector space is graded by µ. It follows from Uhlenbeck’s compactness theorem [28] that the space

Ck = Z2hai

a∈Aflat (Q)/G0(Q) µM(a)=k is ﬁnite dimensional for every integer k. It follows also from Uhlenbeck’s compactness theorem that the moduli space M(a−, a+) consists of ﬁnitely many instantons (modulo time shift) whenever

µ(a−) − µ(a+) = 1.

− + Let n2(a , a ) denote the number of such instantons, counted modulo 2. These numbers determine a linear map ∂ : Ck+1 → Ck deﬁned by

∂hbi = n2(b, a)hai µ(Xa)=k for b ∈ Aﬂat(Q) with µ(b) = k + 1. In [11] Floer proved that (C, ∂) is a chain complex and that its homology is an invariant of the bundle Q → M.

Theorem 2.8 Assume (H2) and (H3). (1) The above map ∂ : C → C satisﬁes ∂2 = 0. The associated homology groups

inst ker ∂k−1 HFk (M; Q) = im∂k are called the Floer homology groups of pair (M, Q).

inst (2) The Floer homology groups HFk (M; Q) are independent of the metric on M used to construct them.

To prove ∂2 = 0 we must show that

n2(c, b)n2(b, a) ∈ 2Z

b∈Aflat (Q)/G0(Q) µX(b)=k whenever µ(a) = k − 1 and µ(c) = k + 1. This involves a glueing argument for pairs of instantons running from c to b and from b to a. Such a pair gives rise to a (unique) 1-parameter family of instantons running from c to a. Now the space of instantons running from c to a (modulo time shift) is a 1-dimensional manifold and has therefore an even number of ends. This implies ∂2 = 0. That any two instanton homology groups corresponding to diﬀerent metrics are naturally isomorphic can be proved along similar lines.

11 inst Remark 2.9 (i) The Floer homology groups HF∗ (M; Q) are new invariants of the 3-manifold M which cannot be derived from the classical invariants of diﬀerential topology. (ii) The Floer homology groups are graded modulo 4. It follows from Lemma 2.3 that there exists a gauge transformation g ∈ G(Q) of degree 1 and by The- orem 2.7 the map a 7→ g∗a induces an isomorphism

inst inst HFk+4(M; Q) = HFk (M; Q). Note however that the even gauge transformations g ∈ Gev(Q) only give inst rise to a grading modulo 8. So the Euler characteristic of HF∗ (M; Q) is the number 3 inst k inst χ(HF∗ (M; Q)) = (−1) dim HFk (M; Q) kX=0 1 = (−1)µ(a). 2 ∈A Gev a flat(XQ)/ (Q) This is the Casson invariant λ(M; Q). (iii) By Lemma 2.3 the group of components of the space of degree-0 gauge transformations {g ∈ G(Q) : deg(g) = 0} is isomorphic to the ﬁnite group

1 Γ = {η(g) : g ∈ G(Q), deg(g) = 0} ⊂ H (M; Z2).

inst This group acts on HFk (M; Q) for every k through permutations of the canonical basis. By Remark 2.6 the group Γ does not act freely. The above definition of the Casson invariant ignores this action of Γ. (iv) The same construction works over the integers. For this we must assign a number +1 or −1 to each instanton running from a− to a+ whenever µ(a−) − µ(a+) = 1. This involves a consistent choice of orientations for the Moduli spaces M(a−, a+). (v) If Hypothesis (H3) is violated then the Floer homology groups can still be defined. The construction then requires a suitable perturbation of the Chern-Simons functional. This will be discussed in Section 7. (vi) It represents a much more serious problem if Hypothesis (H2) fails. This is because of the presence of flat connections which are not regular. Such connections cannot be removed by a gauge invariant perturbation. (vii) If Q is the trivial SO(3)-bundle over a homology-3-sphere M then the only flat connection which is not regular is the trivial connection. In this case the difficulty mentioned in (vi) does not arise. This is in fact the context of Floer’s original work on instanton homology [11].

12 3 Floer homology for symplectic ﬁxed points

In [12] Floer developed a similar theory for fixed points of symplectomorphisms. In his original work Floer assumed that the symplectomorphism φ is exact and symp he proved that the Floer homology groups HF∗ (M, φ) are isomorphic to the homology of the underlying symplectic manifold (M, ω). Like Floer we assume that M is compact and monotone. This means that the first Chern class c1 of T M agrees over π2(M) with the cohomology class of the symplectic form ω. Unlike Floer we assume in addition that M is simply connected but we do not require the symplectomorphism φ : M → M to be isotopic to the identity. Then there are Floer homology groups of φ whose Euler characteristic is the Lefschetz number. Here is how this works. The fixed points of φ can be represented as the critical points of a function on the space of smooth paths

Ωφ = {γ : R → M : γ(s + 1) = φ(γ(s))} .

The tangent space to Ωφ at γ is the space of vector ﬁelds ξ(s) ∈ Tγ(s)M along γ such that ξ(s + 1) = dφ(γ(s))ξ(s). The space Ωφ carries a natural 1-form

Tγ Ωφ → R : ξ 7→ − ι(ξ)ω. Zγ

This 1-form is closed but not exact since Ωφ is not simply connected. Since M is simply connected the fundamental group of Ωφ is π1(Ωφ) = π2(M). The universal cover Ωφ can be explicitly represented as the space of homotopy classes of smooth maps u : R × I → M such that e u(s, 0) = x0, u(s + 1, t) = φ(u(s, t)), u(s, 1) = γ(s).

Here x0 = φ(x0) is a reference point chosen for convenience of the notation. The homotopy class [u] is to be understood subject to the boundary condition at t = 0 and t = 1. The second homotopy group π2(M) acts on Ω˜ φ by taking connected sums. A sphere in M can be represented by a function v : R×I → M such that v(s, 0) = v(s, 1) = v(0, t) = x0 and v(s + 1, t) = v(s, t) for s, t ∈ R. The connected sum of [u] ∈ Ωφ and [v] ∈ π2(M) is given by u#v(s, t) = v(2t, s) for t ≤ 1/2 and u#v(s, t) = u(2t − 1, s) for t ≥ 1/2. The pullback of the abovee 1-form on Ωφ is the diﬀerential of the function

∗ aφ : Ωφ → R, aφ(u) = u ω Z called the symplectic eactione . This functione satisﬁes

∗ aφ(u#v) = aφ(u) + v ω 2 ZS e e 13 for every sphere v : S2 → M. Since M is monotone the symplectic form ω is integral and hence aφ descends to a map aφ : Ωφ → R/Z. By construction the diﬀerential of aφ is the above 1-form on Ωφ. So the critical points of aφ are the constant paths in Ωeφ and hence the ﬁxed points of φ.

aφ - Ωφ R e e π π

? ? aφ - Ωφ R/Z

An almost complex structure J : T M → T M is said to be compatible with ω if the bilinear form hξ, ηi = ω(ξ, Jη) deﬁnes a Riemannian metric on M. The space J (M, ω) of such almost complex structures is contractible. A symplectomorphism φ acts on J (M, ω) by pullback ∗ J 7→ φ J. To construct a metric on Ωφ choose a smooth family Js ∈ J (M, ω) such that ∗ Js = φ Js+1. (5)

This condition guarantees that for any two vectorﬁelds ξ, η ∈ Tγ Ωφ the expres- sion hξ(s), η(s)is = ω(ξ(s), Js(γ(s))η(s)) is of period 1 in s. Hence deﬁne the inner product of ξ and η by

1 hξ, ηi = hξ(s), η(s)is ds. Z0 The gradient of aφ with respect to this metric on Ωφ is given by grad aφ(γ) = 2 Js(γ)γ˙ . Hence a gradient ﬂow line of aφ is a smooth map u : R → M satisfying the nonlinear partial diﬀerential equation ∂u ∂u + J (u) = 0 (6) ∂t s ∂s and the periodicity condition

u(s + 1, t) = φ(u(s, t)). (7)

Condition (5) guarantees that whenever u(s, t) is a solution of (6) then so is v(s, t) = φ(u(s − 1, t)). Condition (7) requires that these two solutions agree.

14 As in section 2 equations (6) and (7) do not define a well posed Cauchy problem and the Morse index of any critical point is infinite. However, the solutions of (6) are precisely Gromov’s pseudoholomorphic curves [16]. It follows from Gromov’s compactness that every solution of (6) and (7) with finite energy ∞ 1 1 2 2 E(u) = |∂su|s + |∂tu|s dsdt < ∞ 2 −∞ 0 Z Z has limits lim u(s, t) = x = φ(x). (8) t→∞ (See for example [24], [31].) Conversely, any solution of (6), (7), and (8) has finite energy. Given any two fixed points x let M(x−, x+) denote the space of these solutions. For any function u : R2 → M which satisfies (7) and (8) we introduce the 2n Maslov index as follows. Let Φ(s, t) : R → Tu(s,t)M be a trivialization of u∗T M as a symplectic vector bundle such that Φ(s + 1, t) = dφ(u(s, t))Φ(s, t). Consider the paths of symplectic matrices Ψ(s) = Φ(s, ∞)−1Φ(0, ∞) ∈ Sp(2n; R). These satisfy Ψ(0) = 1l and Ψ(1) is conjugate to dφ(x). In particular 1 is not an eigenvalue of Ψ(1). The homotopy class of such a path is determined by its Maslov index µ(Ψ) ∈ Z introduced by Conley and Zehn- der [5]. Roughly speaking the Maslov index counts the number of times s such that 1 is an eigenvalue of the symplectic matrix Ψ(s). The integer µ(u) = µ(Ψ−) − µ(Ψ+) is independent of the choice of the trivialization and is called the Maslov index of u. This number depends only on the homotopy class of u. For a detailed account of the Maslov index and its role in Floer homology we refer to [25]. The next theorem is due to Floer [12]. Lemma 3.1 If u satisfies (7) and (8) then

∗ µ(u#v) = µ(u) + 2 v c1 2 ZS for any sphere v : S2 → M. Theorem 3.2 For a generic family of almost complex structures satisfying (5) the space M(x−, x+) is a ﬁnite dimensional manifold for every pair of ﬁxed points x = φ(x). The dimension of M(x−, x+) is given by the Maslov index − + dimu M(x , x ) = µ(u) locally near u ∈ M(x−, x+).

15 By Lemma 3.1 the Maslov index induces a map

µ : Fix(φ) → Z2N such that µ(u) = µ(x−) − µ(x+) (mod 2N) for every solution u of (7) and (8). Here the integer N is defined by c1(π2(M)) = NZ. This function µ : Fix(φ) → Z2N is only defined up to an additive constant. We may choose this constant such that (−1)µ(x) = sign det(1l − dφ(x)) (9) for every fixed point x = φ(x). The manifold M(x−, x+) is not connected and the dimension depends on the component. It follows from Theorem 3.2 that the dimension is well defined modulo 2N

dim M(x−, x+) = µ(x−) − µ(x+) (mod 2N).

The Floer homology groups of φ can now be constructed as follows. Let C be the vector space over Z2 freely generated by the ﬁxed points of φ. This vector space is graded modulo 2N by the Maslov index. It is convenient to deﬁne

Ck = Z2hxi x=φ(x) µ(x)=kM(mod 2N) for every integer k keeping in mind that Ck+2N = Ck. Since M is monotone it follows from Gromov’s compactness for pseudoholomorphic curves that the 1-dimensional part of the space M(x−, x+) consists of ﬁnitely many connecting orbits (modulo time shift) whenever µ(x−) − µ(x+) = 1(mod 2N). Let − + n2(x , x ) be the number of these connecting orbits modulo 2. This gives a linear map ∂ : Ck+1 → Ck via the formula

∂hyi = n2(y, x)hxi x=φ(x) µ(x)=kX(mod 2N) for y = φ(y) with µ(y) = k + 1(mod 2N). In [12] Floer proved that (C, ∂) is a chain complex.

Theorem 3.3 Assume that M is simply connected and monotone. (1) The above map ∂ : C → C satisﬁes ∂2 = 0. The associated homology groups

symp ker ∂k−1 HFk (M; φ) = im∂k are called the Floer homology groups of the pair (M, φ). symp (2) The Floer homology groups HFk (M, φ) are independent of the almost complex structure J used to construct them; they depend on φ only up to symplectic isotopy.

16 (3) If φ is isotopic to the identity in the class of symplectomorphisms then symp HF∗ (M; φ) is naturally isomorphic to the homology of the manifold M symp Z HFk (M; φ) = Hj (M; 2). j=k(moMd 2N) symp Proof of (2): To prove that HF∗ (M; φ) depends only on the isotopy class of φ we must generalize the above construction. Let

−1 φs = ψs ◦ φ be an isotopy of symplectomorphisms from φ0 = φ to φ1. Since M is simply connected there exists a family of Hamiltonian vector ﬁelds Xs : M → T M such that d ψ = X ◦ ψ . ds s s s

This means that the 1-form obtained by contracting Xs with the symplectic form ω is exact ι(Xs)ω = dHs. The isotopy can be chosen such that

ψs+1 ◦ φ1 = φ0 ◦ ψs or equivalently ∗ φ Xs+1 = Xs, Hs+1 ◦ φ = Hs. (10) Now replace equation (6) by

∂u ∂u + J (u) − ∇H (u) = 0. (11) ∂t s ∂s s

Here ∇Hs = JsXs is the gradient of the Hamiltonian function Hs : M → R with respect to the metric induced by Js. It follows from (5) that whenever u(s, t) is a solution of (11) then so is v(s, t) = φ(u(s − 1, t)). Hence the periodicity condition (7) is consistent with (11). One can prove by exactly the same symp arguments as in [12] that HF∗ (M; φ) is isomorphic to the Floer homology constructed with (11) instead of (6). We denote these Floer homology groups symp by HF∗ (M; φ, H). The stationary points of equation (11) (that is ∂u/∂t = 0) with the boundary condition (7) are the paths γ ∈ Ωφ for which γ(s) = ψs(γ(0)). These paths are −1 in one-to-one correspondence with the ﬁxed points of φ1 = ψ1 φ. Now let u : R2 → M be any solution of (11) and (7) and deﬁne

−1 ∗ v(s, t) = ψs (u(s, t)), Is = ψs Js.

17 ∗ Then φ1 Is+1 = Is and v(s, t) satisﬁes the partial diﬀerential equation ∂v ∂v + I (v) = 0 (12) ∂t s ∂s and the periodicity condition

v(s + 1, t) = φ1(v(s, t)). (13)

Hence there is a one-to-one correspondence between the solutions of (11) and (7) on the one hand and the solutions of (12) and (13) on the other hand. This symp symp shows that HF∗ (M; φ, H) is isomorphic to HF∗ (M; φ1). 2

Remark 3.4 (i) A similar construction works for monotone symplectic manifolds M which are not simply connected. In this case Ωφ will no longer be connected and there are Floer homology groups for every homotopy class of paths. Moreover, the fundamental group of Ωφ will no longer be isomor- 1 phic to π2(M) and aφ may not take values in S . If this is the case then the Floer homology groups will be modules over a suitable Novikov ring as in [17]. Finally, not every isotopy of symplectomorphisms corresponds to a time dependent Hamiltonian vector field but the Floer homology groups will only be invariant under Hamiltonian isotopy. (ii) In [17] the construction of the Floer homology groups has been generalized to some classes of non-monotone symplectic manifolds. The results in [17] include the case where the first Chern class c1 vanishes over π2(M) but ω does not. In this case the Floer homology groups are modules over the Novikov ring associated to the ordering on π2(M) determined by ω. (iii) Theorem 3.3 implies the Arnold conjecture for simply connected monotone symplectic manifolds M: If φ is the time-1-map of a time dependent Hamiltonian vector field with nondegenerate fixed points then the number of fixed points of φ can be estimated below by the sum of the Betti numbers. In [12] Floer proved this result without assuming M to be simply connected.

symp (iv) It follows from (9) that the Euler characteristic of HF∗ (M; φ) is the Lefschetz number of φ

2N−1 symp k symp χ(HFk (M; φ)) = (−1) dim HFk (M; φ) kX=0 = sign det(1l − dφ(x)) x=Xφ(x) = L(φ).

18 4 Flat connections over a Riemann surface

Let π : P → Σ be principal SO(3)-bundle over a compact oriented Riemann surface Σ of genus k. Up to isomorphism there are only two such bundles characterized by the second Stiefel-Whitney class. We assume w2(P ) = 1 so the bundle is nontrivial. As in section 1 let A(P ) denote the space of connections on P and G(P ) denote the group of gauge transformations. The component of the identity G0(P ) = {g ∈ G(P ) : g ∼ 1} can be characterized as the subgroup of even gauge transformation that is those which lift to SU(2). Alternatively G0(P ) can be described as the kernel of the 1 epimorphism η : G(P ) → H (Σ; Z2) which, as in section 1, assigns to each gauge transformation its parity η(g). Here η induces an isomorphism from the group 1 2k of components of G(P ) to H (Σ; Z2) ≈ Z2 Recall from [2] that the aﬃne space A(P ) is an inﬁnite dimensional symplectic manifold with symplectic form

ωA(a, b) = ha ∧ bi (14) ZΣ 1 for a, b ∈ TAA(P ) = Ω (gP ). The gauge group G0(P ) acts on this manifold by 0 symplectomorphisms. The Lie algebra Lie(G0(P )) = Ω (gP ) acts by Hamilto- 1 0 nian vector ﬁelds A(P ) → Ω (gP ) : A 7→ dAξ where ξ ∈ Ω (gQ). The associated R Hamiltonian functions are A(P ) → : A 7→ ΣhFA ∧ ξi. Thus the curvature

2 R A(P ) 7→ Ω (gP ) : A 7→ FA is the moment map and the corresponding Marsden-Weinstein quotient is the moduli space M(P ) = Aflat(P )/G0(P ). of flat connections modulo even gauge equivalence. Since P is the nontrivial SO(3)-bundle the space M(P ) is a compact manifold of dimension 6k − 6 provided that k ≥ 2. Moreover, every conformal structure on Σ induces a Kähler structure on M(P ). To see this consider the DeRham complex 0 dA 1 dA 2 Ω (gP ) −→ Ω (gP ) −→ Ω (gP ) twisted by a flat connection A. It follows from Lemma 4.1 below that dA : 0 1 0 1 Ω → Ω is injective and hence HA(Σ) = 0. The first cohomology HA(Σ) = ∗ 1 ker dA ∩ dA ⊂ Ω (gP ) appears as the tangent space of the manifold M(P ) at A. A conformal structure on Σ determines a Hodge-∗-operator

1 1 HA(Σ) → HA(Σ) : a 7→ ∗a

19 for A ∈ Aflat(P ). These operators form an integrable complex structure on M(P ) which is compatible with the symplectic form (14) The associated Kähler metric is given by ha, bi = ha ∧ ∗bi ZΣ 1 for a, b ∈ HA(Σ). Via the holonomy the space M(P ) can be identified with the space of odd representations of the fundamental group of P in SU(2):

Homodd(π (P ), SU(2)) M(P ) = 1 . SU(2) More precisely, since g = so(3) = su(2) the holonomy of a ﬂat connection A ∈ Aﬂat(P ) at a point p0 ∈ P determines a homomorphism ρA : π1(P ) → SU(2) whose image is denoted by HA(p0) = ρA(π1(P )). Since P is the nontrivial bundle its fundamental group is given by 2k + 1 generators

α1, . . . αk, β1, . . . , βk, ε with relations

k 2 [αj , βj ] = ε, [αj , ε] = 1, [βj , ε] = 1, ε = 1. j=1 Y

A homomorphism ρ : π1(P ) → SU(2) is called odd if ρ(ε) = −1. Any such homomorphism is given by a 2k-tuple of matrices Uj, Vj ∈ SU(2) such that

k [Uj , Vj ] = −1 j=1 Y The space of conjugacy classes of such homomorphisms is easily seen to be a compact manifold of dimension 6k − 6. In particular, every ﬂat connection on P is regular.

Lemma 4.1 If A ∈ Aﬂat(P ) then GA ∩ G0(P ) = {1}.

Proof: Let g ∈ GA ∩ G0(P ) and let g˜ : P → SU(2) be a lift of g. Then g˜(p0) commutes with HA(p0). By the above discussion HA(p0) is not an abelian subgroup of SU(2). Hence g˜(p0) = 1l and g(p0) = 1l. 2 The topology of the moduli space M(P ) has been studied extensively by Atiyah and Bott [2] and Newsteadt [21]. We recall those results which are of interest to us.

Theorem 4.2 Assume k ≥ 2. The moduli space M(P ) is connected and simply connected and π2(M(P )) = Z.

20 It is easy to see that M(P ) is connected for k ≥ 1 and simply connected for k ≥ 2 [21]. Deﬁne

k 2k Rk : SU(2) → SU(2) : (U1, . . . , Uk, V1, . . . , Vk) 7→ [Uj , Vj ]. j=1 Y 2k A 2k-tuple in Xk = SU(2) is a singular point for Rk if and only if all 2k matrices commute. Hence the set Sk of singular points is a (2k +2)-dimensional −1 stratified subvariety of Zk = Rk (1l). Hence there is a fibration Rk : Xk \Zk → −1 SU(2) \ {1l} with fibre Fk = Rk (−1l) over a contractible base. The space Xk \ Zk is connected for k ≥ 1 and simply connected for k ≥ 2 and so is Fk. Now SO(3) acts freely on Fk and the quotient is M(P ). The homotopy exact sequence of the fibration

SO(3) ,→ Fk → M(P ) shows that M(P ) is connected and simply connected. Using the linking number of a sphere in Xk \ Zk with the codimension-3 submanifold Zk \ Sk we see that π2(Fk) = π2(Xk \ Zk) = Z. But in this case the fundamental group of SO(3) enters the exact sequence

0 → Z → π2(M(P )) → Z2 → 0 and we can only deduce that either π2(M(P )) = Z or π2(M(P )) = Z ⊕ Z2. We must rule out the latter case. This requires the approach of Atiyah and Bott [2] using inﬁnite dimensional Morse theory.

Theorem 4.3 Assume k ≥ 2. Then the (infinite dimensional) space Aflat(P ) of flat connections on P is simply connected and π2(Aflat(P )) = 0. The proof of this result is based on Morse theory for the Yang-Mills functional 2 YM(A) = kFAk ZΣ 1 on the space of connections A(P ). The idea is to extend a loop A1 : S → Aflat(P ) to a map A : D → A(P ) on the unit disc. Then use the gradient flow of the Yang-Mills functional to ‘push this extension down’ to the set Aflat(P ) of absolute minima. This requires that all the non-minimal critical manifolds have Morse index at least 3 so that the extension A in general position does not intersect their stable manifolds. The same consideration gives π2(Aflat(P )) = 0 if these Morse indices are at least 4. The details of this argument are carried out in [6]. In [2] Atiyah and Bott used an alternative stratification of the space A(P ) to prove Theorem 4.3. Now the homotopy exact sequence of the fibration G0 ,→ Aflat(P ) → M(P ) shows that π2(M(P )) = π1(G0(P )) = Z.

21 The last identity follows from Lemma 1.2 applied to the bundle Q = P × S1 1 over the 3-manifold M = Σ × S . The isomorphism π1(G0(P )) = Z is given by 1 the degree of a loop g(s) ∈ G0(P ) regarded as a gauge transformation of P ×S . We shall now give a more explicit description of the isomorphism from π1(G0(P )) to π2(M(P )). Let A0 ∈ Aflat(P ) be a flat connection and let g(θ) ∈ G0(P ) be a loop of gauge transformations such that g(0) = g(1) = 1l. By Theorem 4.3 there exists a map A : D → Aflat(P ) on the unit disc such that

2πiθ ∗ A(e ) = A1(θ) = g(θ) A0. (15)

This map represents a sphere in the moduli space M(P ) since the boundary of D is mapped to a point. An easy calculation shows that the integral of the symplectic form over this sphere is given by

∂A ∂A 1 1 dA ∧ dsdt = 1 ∧ (A − A ) dθ ∂s ∂t 2 dθ 1 0 ZD ZΣ Z0 ZΣ = 8π2 deg(g). (16)

The last identity follows from equation (2) applied to the Chern-Simons functional on the bundle Q = P × S1.

2k Remark 4.4 (i) The group π0(G(P )) = G(P )/G0(P ) = Z2 of components of the gauge group acts on the moduli space by symplectomorphism M(P ) → M(P ) : [A] 7→ [g∗A]. This action is not free: it is easy to construct a flat connection whose isotropy subgroup GA is nontrivial but discrete. (ii) The tangent bundle of M(P ) is a complex vector bundle and therefore has Chern classes. It follows from our theorem about the spectral flow [7] that 2 the integral of the first Chern class c1 ∈ H (M(P ); Z) over the sphere A : D → Aflat(P ) satisfying (15) is given by 2 deg(g). This was already known to Atiyah and Bott [2].

5 Mapping cylinders

Continue the notation of the previous section. Any diffeomorphism h : Σ → Σ lifts to an automorphism f : P → P since h∗P is isomorphic to P . We assume throughout that h is orientation preserving. The automorphism f induces a symplectomorphism φf : M(P ) → M(P ) ∗ defined by φf ([A]) = [f A]. In the context of representations of the fundamental group this symplectomorphism is given by ρ 7→ ρ ◦ f∗. Hence the symplectomorphism φf depends only on the homotopy class of f. A fixed point of φf is

22 ∗ ∗ an equivalence class of a ﬂat connection A0 ∈ Aﬂat(P ) such that g f A0 = A0 for some gauge transformation g ∈ G0(P ). This can be written as

∗ fg A0 = A0 where the automorphism fg ∈ Aut(P ) is deﬁned by

fg(p) = f(p)g(p) for p ∈ P . The diﬀerential of φf at the ﬁxed point A0 is the linear map

∗ 1 1 fg : HA0 (Σ) → HA0 (Σ). 1 Here we identify HA with the quotient ker dA0 /im dA0 rather than the space of 0 ∗ harmonic forms which will in general not be invariant under fg . Remark 5.1 For every g ∈ G(P ) and every A ∈ A(P )

f ∗g∗A = (g ◦ f)∗f ∗A.

Hence the symplectomorphism M(P ) → M(P ) : [A] 7→ [g∗A] commutes with φf whenever g is homotopic to g ◦ f or equivalently the parity η(g) is in the kernel of the homomorphism 1l − h∗ : H1(Σ, Z2) → H1(Σ, Z2). Deﬁne

Gf (P ) = {g ∈ G(P ) : g ∼ g ◦ f} , Γf = Gf (P )/G0(P ).

∗ Then Γf ' ker(1l − h ) is a finite group which acts on M(P ) by symplectomorphisms which commute with φf . In particular, Γf acts on the fixed points of 0 ∗ φf . More explicitly, if [A0] ∈ Fix(φf ) and g0 ∈ Gf (P ) then [A0] = [g0 A0] ∈ ∗ ∗ Fix(φf ). To see this choose g ∈ G0(P ) such that g f A0 = A0 and define 0 −1 0∗ ∗ 0 0 g = (g0 ◦ f) gg0 ∈ G0(P ). Then g f A0 = A0. The automorphism f also determines a principal SO(3)-bundle

Pf → Σh.

Here Q = Pf denotes the mapping cylinder of P for the automorphism f. That is the set of equivalence classes of pairs [p, s] ∈ P × R under the equivalence relation generated by [p, s + 1] ≡ [f(p), s]. Likewise the 3-manifold M = Σh denotes the mapping cylinder of the Riemann surface Σ for the diﬀeomorphism h. This bundle satisﬁes hypothesis (H1) of section 1. A connection a ∈ A(Pf ) 0 is a 1-form a = A + Φ ds where A(s) ∈ A(P ), Φ(s) ∈ Ω (gP ) for s ∈ R and A(s + 1) = f ∗A(s), Φ(s + 1) = Φ(s) ◦ f. (17)

The group G(Pf ) of gauge transformations of Pf consists of smooth 1-parameter families of gauge transformations g(s) ∈ G(P ) such that

g(s + 1) = g(s) ◦ f.

23 Such a gauge transformation acts on a connection a = A + Φ ds ∈ A(Pf ) by

g∗a = g∗A + g−1g˙ + g−1Φg ds.

∗ ∗ Here we use the notation g ambiguously: g a denotes the action of g ∈ G(Pf ) ∗ on a ∈ A(Pf ) whereas g A denotes the pointwise action of g(s) ∈ G(P ) on A(s) ∈ A(P ).

Remark 5.2 Consider the normal subgroups

G0(Pf ) ⊂ GΣ(Pf ) ⊂ G(Pf ) where GΣ(Pf ) = {g ∈ G(Pf ) : g(s) ∈ G0(Σ) ∀ s} and G0(Pf ) is the component of 1l. Then G(Pf )/GΣ(Pf ) ' Γf , GΣ(Pf )/G0(Pf ) ' Z. The second isomorphism is given by the degree while the ﬁrst follows from the fact that g0 ∈ G(P ) extends to a gauge transformation of Pf if and only if g0 ∈ Gf (P ). The ﬁrst isomorphism shows that Γf is the group of components of gauge transformations of Pf of degree zero

The path space Ωφf can be naturally identiﬁed with a subquotient of the space of connections on Pf . Consider the subspace

∗ AΣ(Pf ) = {A + Φ ds ∈ A(Pf ) : FA = 0, dA (dA/ds − dAΦ) = 0} .

If A + Φ ds ∈ AΣ(Pf ) then Φ is uniquely determined by A. In fact dA/ds − 1 dAΦ represents the projection of dA/ds onto the space HA(Σ) = T[A]M(P ) of harmonic forms.

Proposition 5.3 There are natural bijections

Ωφf ' AΣ(Pf )/GΣ(Pf ), Ωφf ' AΣ(Pf )/G0(Pf ).

In particular, Ωφf is connected and e

π1(Ωφf ) ' GΣ(Pf )/G0(Pf ) ' π2(M(P )). R Proof: A point in the space Ωφf is a smooth path γ : → M(P ) such that γ(s + 1) = φf (γ(s)). Such a path lifts to a smooth map A : R → Aﬂat(P ) satisfying A(s + 1) = h(s)∗f ∗A(s). 0 0 for some smooth map h : R → G0(P ). Two such pairs (A, h) and (A , h ) represent the same path γ : R → M(P ) if and only if there exists a smooth map g : R → G0(P ) such that

− A0(s) = g(s)∗A(s), h0(s) = (g(s) ◦ f) 1 h(s)g(s + 1).

24 Every path γ can be represented by a pair (A, h) with h(s) ≡ 1. To see this choose, for any given pair (A, h), a path g : [0, 1] → G0(P ) such that g(s) = 1 for s near 0 and g(s) = h(s)−1 for s near 1. Then g(s) extends to a unique function R → G0(P ) satisfying

h(s)g(s + 1) = g(s) ◦ f.

This proves the first statement. To prove the second statement recall from Theorem 4.3 that Aflat(P ) is connected and simply connected and π2(Aflat(P )) = 0. Hence AΣ(Pf ) is connected and simply connected. By the homotopy exact sequence

π1(AΣ(Pf )) → π1(AΣ(Pf )/G0(Pf )) → π0(G0(Pf )) the quotient AΣ(Pf )/G0(Pf ) is simply connected. Hence this quotient is the 2 universal cover of AΣ(Pf )/GΣ(Pf ) ' Ωφf .

The Chern-Simons functional on the space of connections A(Pf ) is given by the formula 1 1 CS(A + Φ ds) = hA˙ ∧ (A − A0)i + hFA ∧ Φi ds 2 0 Σ Z Z ∗ Here A0 is any flat connection on P such that f A0 = A0. A simple calculation shows that the restriction of CS to AΣ(Pf ) induces the symplectic action functional on Ωφf = AΣ(Pf )/G0(Pf ). We shall in fact prove that the critical points of the Chern-Simons functional, that is the flat connections on Pf , agree with the criticale points of the symplectic action, that is the fixed points of φf . To see this note that the curvature of the connection a = A + Φ ds is the 2-form

FA+Φ ds = FA + dAΦ − A˙ ∧ ds Thus the flat connections on Pf are smooth families of flat connections A(s) ∈ Aflat(P ) such that dA/ds ∈ im dA and (17) is satisfied.

Proposition 5.4 The map A(Pf ) → A(P ) : A + Φ ds 7→ A(0) induces a bijec- tion Aﬂat(Pf )/GΣ(Pf ) ' Fix(φf ).

In particular Aﬂat(Pf )/G(Pf ) ' Fix(φf )/Γf .

Proof: First assume that a = A + Φ ds ∈ Aflat(Pf ). Then A(s) ∈ Aflat(P ) for every s and A˙ = dAΦ. Let g(s) ∈ G(P ) be the unique solution of the ordinary ∗ differential equation g˙ + Φg = 0 with g(0) = 1l. Then g(s) A(s) ≡ A0 and ∗ ∗ ∗ hence it follows from (17) that g(1) f A0 = g(1) A(1) = A0. By construction, g(1) ∈ G0(P ) and hence A0 represents a fixed point of φf . This proves that there is a well defined map Aflat(Pf )/GΣ(Pf ) → Fix(φf ) given by A + Φ ds 7→ A(0).

25 We prove that this map is onto. Suppose that A0 ∈ Aflat(P ) represents a ∗ ∗ fixed point of φf and let g1 ∈ G0(P ) such that g1f A0 = A0. Choose a smooth 1-parameter family of connections g(s) ∈ G0(P ) such that g(0) = 1, g(1) = g1 and g(s + 1) = (g(s) ◦ f)g1 for every s. Let A(s) ∈ Aflat(P ) be defined by ∗ −1 g(s) A(s) = A0 and Φ(s) = −g˙(s)g(s) . Then a = A + Φ ds is the required flat connection on Pf . 0 0 We prove that the map is injective. Let a, a ∈ Aflat(Pf ) such that A (0) = ∗ g0 A(0) for some g0 ∈ G0(P ). Define g(s) ∈ G0(P ) to be the unique solution of 0 the ordinary differential equation g˙ = gΦ − Φg, g(0) = g0. Then

d ∗ −1 −1 −1 0 −1 0 g A = g A˙g + d ∗ (g g˙) = g d Φg + d ∗ (Φ − g Φg) = d ∗ Φ . ds g A A g A g A

0 ∗ 0 0 Here we have used A˙ = dAΦ. Since A (0) = g(0) A(0) and A˙ = dA0 Φ it follows that g(s)∗A(s) = A0(s) for every s. Moreover it follows from (17) that −1 g(s+1) g(s)◦f ∈ GA(s+1)∩G0(P ). By Lemma 4.1 this implies g(s+1) = g(s)◦f. 0 ∗ Hence g(s) deﬁnes a gauge transformation of Pf and a = g a. 2

Proposition 5.5 A flat connection A + Φ ds ∈ Aflat(Pf ) is nondegenerate as a critical point of the Chern-Simons functional if and only if [A(0)] is a nondegenerate fixed point of φf .

Proof: Let a = A + Φ ds ∈ Aflat(Pf ) be a flat connection and define A0 = ∗ A(0) ∈ Aflat(P ). Then fg A0 = A0 where g ∈ G0(P ) is defined by g = g(1) for the unique solution g(s) ∈ G0(P ) of g˙(s) + Φ(s)g(s) = 0 with g(0) = 1l. We must show that ∗ 1 1 1l − fg : HA0 (Σ) → HA0 (Σ) 1 is an isomorphism if and only if HA+Φ ds(Σh) = 0. The latter means that 1 whenever α + φ ds ∈ Ω (gPf ) is an infinitesimal connection such that

dAα = 0, dAφ = α˙ + [Φ ∧ α] (18)

0 then there exists a ξ ∈ Ω (gPf ) such that

α = dAξ, φ = ξ˙ + [Φ ∧ ξ]. (19)

∗ Replacing a by g a and f by fg(1) we may assume without loss of generality that Φ(s) ≡ 0 and A(s) ≡ A0. ∗ 1 1 g Assume ﬁrst that 1l − f is an isomorphism of HA0 . Let α(s) ∈ Ω ( P ) 0 ∗ and φ ∈ Ω (gP ) satisfy (18) and the boundary conditions α(s + 1) = f α(s), φ(s + 1) = φ(s) ◦ f. Then

1 ∗ f α(0) − α(0) = dA0 φ(s) ds, dA0 α(0) = 0. Z0

26 ∗ 1 1 0 g Since 1l − f : HA0 → HA0 is injective there exists a ξ0 ∈ Ω ( P ) such that

α(0) = dA0 ξ0. Deﬁne s ξ(s) = ξ0 + φ(θ) dθ. Z0 ˙ ˙ Then φ = ξ, hence α˙ = dA0 ξ, and hence α = dA0 ξ. The latter identity implies ∗ that dA0 ξ(s + 1) = f dA0 ξ(s) and hence ξ(s + 1) = ξ(s) ◦ f. ∗ 1 1 Conversely, suppose that 1l − f : HA0 → HA0 is not injective. Then there 1 0 exist α0 ∈ Ω (gP ) and ξ0 ∈ Ω (gP ) such that

∗ f α0 − α0 = dA0 ξ0, dA0 α0 = 0, α0 ∈/ imdA0 .

0 Choose any function φ(s) ∈ Ω (gP ) satisfying

1 ξ0 = φ(s) ds, φ(s + 1) = φ(s) ◦ f. Z0 For example take a cutoﬀ function β : [0, 1] → R of mean value 1 which vanishes j near s = 0 and s = 1 and deﬁne φ(s + j) = β(s)ξ0 ◦ f for 0 ≤ s ≤ 1 and j ∈ Z. 1 Let α(s) ∈ Ω (gP ) be the unique solution of α˙ = dA0 φ with α(0) = α0. Then α and φ satisfy (18) but are not of the form (19) since α0 ∈/ imdA0 . This shows 1 2 that HA+Φ ds 6= 0.

Remark 5.6 It is an open question whether every symplectomorphism φ : M(P ) → M(P ) is isotopic (within the group of symplectomorphisms) to one of the form φf for f ∈ Aut(P ). It is also an open question whether f0 and f1 are isotopic whenever φf0 and φf1 are isotopic.

6 Instantons and holomorphic curves

The proof of Theorem 1.1 is based on a comparison between holomorphic curves in M(P ) and self-dual instantons on the 4-manifold Σh × R. To carry this out we must choose a metric on Σh. Let h , is be a one parameter family of metrics on Σ such that hdh(z)ζ0, dh(z)ζ1is = hζ0, ζ1is+1. j 2−j Then the associated Hodge-∗-operators ∗s : Ω (gP ) → Ω (gP ) satisfy

∗ ∗ ∗s+1 ◦ f = f ◦ ∗s. (20)

j ∗ This deﬁnes a metric on Pf : whenever ξ(s), η(s) ∈ Ω (gP ) with ξ(s+1) = f ξ(s) and η(s + 1) = f ∗η(s) then the function

hξ(s), η(s)is = hξ(s) ∧ ∗sη(s)i ZΣ

27 is of period 1 in s. The inner product of ξ and η is deﬁned as the integral of this function over the unit interval 1 hξ, ηi = hξ(s) ∧ ∗sη(s)i ds. Z0 ZΣ 1 Now recall that the tangent space to M(P ) at A is the quotient HA(Σ) = ker dA/imdA. The metrics on Σ determine a family of complex structures

1 1 Js = ∗s : HA(Σ) → HA(Σ) and condition (20) implies that these satisfy (5) with φ = φf . In order for the 2 space of harmonic forms to be invariant under ∗s we must use the L -adjoint ∗ dA = − ∗s dA∗s with respect to the s-metric. Any smooth function u : R2 → M(P ) lifts to a smooth function A : R2 → Aflat(P ). For any such map the partial derivatives ∂A/∂s and ∂A/∂t lie in the kernel of dA but will in general not be harmonic. To apply a complex structure we must first project these derivatives into the space of harmonic forms corresponding to this complex structure. These projections can be described as 0 ∂A/∂s − dAΦ and ∂A/∂t − dAΨ where Φ, Ψ ∈ Ω (gP ) are uniquely determined by the requirement ∂A ∂A d ∗ − d Φ = 0, d ∗ − d Ψ = 0. A s ∂s A A s ∂t A Thus our function u = [A] : R2 → M(P ) satisfies the nonlinear Cauchy- 2 0 Riemann equations (6) if and only if there exist functions Φ, Ψ : R → Ω (gP ) such that ∂A ∂A − d Ψ + ∗ − d Φ = 0. (21) ∂t A s ∂s A Moreover, the periodicity condition (7) with φ = φf is equivalent to

A(s+1, t) = f ∗A(s, t), Φ(s+1, t) = Φ(s, t)◦f, Ψ(s+1, t) = Φ(s, t)◦f. (22)

The limit condition (8) takes the form

lim A(s, t) = A(s), lim Φ(s, t) = Φ(s), lim Ψ(s, t) = 0 (23) t→∞ t→∞ t→∞ ˙ where A (s) ∈ Aflat(P ) and A = dA Φ . This means that A + Φ ds are flat connections on Pf . Strictly speaking, the periodicity conditions (22) need only be satisfied up to even gauge equivalence. However, any such triple A, Φ, Ψ can be transformed so as to obtain (22). (See the proof of Proposition 4.1.) If two solutions of (21) and (22) are gauge equivalent by a family of even gauge transformations g(s, t) ∈ G0(P ) then

g(s + 1, t) = g(s, t) ◦ f.

28 This means that g deﬁnes a gauge transformation on the bundle Pf ×R over the 4-manifold Σh × R. Moreover, the action of g on the triple A, Φ, Ψ corresponds to the interpretation of this triple as a connection A + Φ ds + Ψ dt on Pf × R. The curvature of this connection is the 2-form ∂A ∂A F = F − − d Φ ∧ ds − − d Ψ ∧ dt A+Φ ds+Ψ dt A ∂s A ∂t A ∂Ψ ∂Φ + − + [Φ, Ψ] ds ∧ dt. ∂s ∂t Hence the connection A + Φ ds + Ψ dt is self-dual if and only if

∂A ∂A − d Ψ + ∗ − d Φ = 0, ∂t A s ∂s A (24) ∂Φ ∂Ψ − − [Φ, Ψ] + ∗ F = 0. ∂t ∂s s A Note that the first equation in (24) agrees with (21) whereas the second equation replaces the condition on A(s, t) to be flat. Now the holomorphic curves described by equation (21) with FA = 0 can be viewed as a limit case of the instantons described by equation (24). Following Atiyah [1] we stretch the mapping cylinder Σh so that the period converges to infinity. Formally this means that equation (22) is replaced by

A(s+1/ε, t) = f ∗A(s, t), Φ(s+1/ε, t) = Φ(s, t)◦f, Ψ(s+1/ε, t) = Φ(s, t)◦f. and in (24) ∗s is replaced by ∗εs. Now rescale A, Φ, and Ψ:

Aε(s, t) = A(s/ε, t/ε), Φε(s, t) = 1/ε Φ(s/ε, t/ε), Ψε(s, t) = 1/ε Ψ(s/ε, t/ε).

The triple Aε, Φε, Ψε then satisﬁes the periodicity condition (22). Moreover, equation (24) becomes

ε ε ∂A ε ∂A ε − d ε Ψ + ∗ − d ε Φ = 0, ∂t A s ∂s A (25) ε ε ∂Φ ∂Ψ ε ε 1 − − [Φ , Ψ ] + ∗ F ε = 0. ∂t ∂s ε2 s A This is equivalent to conformally rescaling the metric on Σ by the factor ε2. It follows from an implicit function theorem that near every solution of (21), (22) and (23) there is a solution of (25), (22) and (23) provided that ε > 0 is suﬃciently small. This is a singular perturbation theorem and care must be taken with the dependence of the linearized operators on ε. Conversely, a

29 family of solutions Aε, Φε, Ψε of (25), (22) and (23) converges as ε tends to 0 and the curvature of Aε converges to 0. The key point is an energy estimate. The Yang-Mills action of the rescaled equation is given by

ε ε ε YMε(A + Φ ds + Ψ dt) ∞ 1 ε 2 ∂A ε 1 2 = − d ε Ψ + kF ε k dsdt (26) ∂t A ε2 A s Z−∞ Z0 s !

= CS(A− + Φ − ds) − CS(A+ + Φ+ ds).

2 This shows that the L -norm of FAε on Σh × R converges to 0 as ε tends to 0. Now Uhlenbeck’s compactness theorem requires an L∞-estimate of the form

1 ε ε sup kFAε k ∞ ×R + k∂A /∂t − dAε Ψ k ∞ ×R < ∞. (27) ε2 L (Σh ) L (Σh ) ε>0 The proof of this estimate involves a bubbling argument. Roughly speaking, the estimate (27) may be violated in arbitrarily small neighborhoods of ﬁnitely many points and in this case either instantons on S4 or instantons on Σ × C or holomorphic spheres in M(P ) will split oﬀ. But this can be avoided in the case which is relevant for the construction of the Floer homology groups namely when the relative Morse index is 1:

µ(A− + Φ− ds) − µ(A+ + Φ+ ds) = 1.

Now there are two such relative Morse indices; one in the Chern-Simons theory given by the spectral ﬂow of the operator family Da(t) (section 2) and one in symplectic Floer homology given by the Maslov index (section 3). We must prove that both relative Morse indices agree. We shall address this problem as well as singular perturbation and compactness in a separate paper.

7 Perturbations

The methods we have discussed so far require the assumption that all flat connections on the bundle Q = Pf respectively all fixed points of the symplectomorphism φf on M(P ) are nondegenerate. The purpose of this section is to show why this assumption is redundant. In particular we wish to apply Theo- rem 1.1 to f = id in which case all flat connections are degenerate. Nevertheless we obtain the following

Theorem 7.1 The instanton homology of the bundle P × S1 over Σ × S1 is naturally isomorphic to the homology of the moduli space MF (P )

inst 1 1 Z HFk (Σ × S , P × S ) = Hj (MF (P ); 2). j=k(moMd 4)

30 To construct the symplectic Floer homology groups in the degenerate case we must perturb the nonlinear Cauchy-Riemann equations (6) by a Hamiltonian term as in the proof of Theorem 3.3. Likewise we must consider perturbations of the Chern-Simons functional on the space of connections on Pf as in [11] and [27] to construct the instanton homology groups of section 2. We shall construct a smooth family of Hamiltonian functions Hs : A(P ) → R which satisfy ∗ ∗ Hs(g A) = Hs(A) = Hs+1(f A) (28) for A ∈ A(P ) and g ∈ G0(P ). The diﬀerential of Hs can be represented by a 1 smooth map Xs : A(P ) → Ω (gP ) such that

dHs(A)α = hXs(A) ∧ αi. ZΣ 1 In other words Xs : A(P ) → Ω (gP ) is the Hamiltonian vector ﬁeld on A(P ) corresponding to the Hamiltonian function Hs. Since Hs is invariant under G0(P ) the vector ﬁelds Xs satisfy

∗ −1 ∗ ∗ Xs(g A) = g Xs(A)g, dAXs(A) = 0, Xs+1(f A) = f Xs(A) (29) for g ∈ G0(P ) and A ∈ A(P ). The vector fields Xs that arise from the holonomy will be smooth with respect to the W k,p-norm for all k and p and hence give rise to a Hamiltonian flow ψs : A(P ) → A(P ) defined by d ψ = X ◦ ψ , ψ = id. ds s s s 0

The diﬀeomorphisms ψs preserve the symplectic structure and are equivariant under the action of G0(P )

∗ ∗ ψs(g A) = g ψs(A), Fψs(A) = FA.

Moreover, −1 ∗ ∗ ψs+1 ◦ ψ1 (f A) = f ψs(A) for A ∈ A(P ). The restriction of this identity to the Marsden-Weinstein quotient M(P ) can be written in the form

−1 ψs+1 ◦ φf,H = φf ◦ ψs, φf,H = ψ1 ◦ φf .

The symplectomorphism φf,H : M(P ) → M(P ) is related to φf by a Hamilto- nian isotopy. To construct the Hamiltonian functions Hs and the Hamiltonian vector ﬁelds Xs we ﬁrst recall some basic facts about the holonomy of a connection on P . For any loop γ(θ + 1) = γ(θ) ∈ P the holonomy determines a map ρ = ργ : A(P ) →

31 SU(2) defined by ρ(A) = g(1) where g(θ) ∈ SU(2) is the unique solution of the ordinary differential equation g˙ + A(γ˙ )g = 0, g(0) = 1. This equation is meaningful since the Lie algebra of SO(3) agrees with the Lie algebra of SU(2). The differential of ρ can be expressed in terms of g: 1 ρ(A)−1dρ(A)α = − g−1α(γ˙ )g dθ Z0 1 for α ∈ Ω (gP ). Now choose 2k embeddings γj : R/Z × R → P of the annulus such that the projections π ◦ γj are orientation preserving, generate the fundamental group of 2k Σ, and satisfy γj (0, λ) = pλ for every j. Denote by ρλ : A(P ) → SU(2) the holonomy along the loops θ 7→ γj (θ, λ) for λ ∈ R. Now choose a smooth family 2k of functions hs : SU(2) → R which are invariant under conjugacy and vanishes for s near 0 and 1. Let β : R → R be a smooth cutoff function supported in [−1, 1] with mean value 1 and define Hs : A(P ) → R by 1 Hs(A) = β(λ)hs(ρλ(A)) dλ Z−1 for A ∈ A(P ) and 0 ≤ s ≤ 1. The partial derivative of hs with respect to Uj can be represented by a 2k function ηj : [0, 1] × SU(2) → su(2) such that

∂hs (U)Uj ξ = hηj (s, U), ξi. ∂Uj 1 Deﬁne Xs : A(P ) → Ω (gP ) by

2k Xs(A) = Xj (s, A) j=1 X 1 1 for 0 ≤ s ≤ 1 where Xj (s, A) ∈ Ω (gP ) is supported in γj (S × [−1, 1]) and ∗ −1 γj Xj (s, A) = βgj ηj (s, ρλ(A))gj dλ.

Here θ 7→ gj(θ, λ) is the holonomy of A along the loop θ 7→ γj (θ, λ). The vector ﬁeld Xs is related to the Hamiltonian Hs as above for 0 ≤ s ≤ 1. Both can be extended to s ∈ R by (28) and (29).

Remark 7.2 Any smooth Hamiltonian function Hs : M(P ) → R can be rep- 2k resented in the form Hs = hs ◦ ρλ : Aflat(P ) → R where hs : SU(2) → R is 2k invariant under conjugacy. The functions hs : SU(2) → R can be chosen such that φf,H has only nondegenerate fixed points. We do not assume here that Hs is invariant under the action of Γf . This would require a transversality theorem which takes account of the action of a finite group.

32 The perturbed Chern-Simons functional CSH : A(Pf ) → R is deﬁned by

1 CSH (a) = CS(a) − Hs(A(s)) ds. Z0 for a = A + Φ ds ∈ A(Pf ). Its diﬀerential is given by

1 dCS(A + Φ ds)(α + φ ds) = h(A˙ − dAΦ − Xs(A)) ∧ αi + hFA ∧ φi ds. 0 Σ Z Z Hence a connection A + Φ ds on Pf is a critical point of CSH if and only if A(s) is ﬂat for every s and A˙ − dAΦ − Xs(A) = 0. (30)

Now the restriction of Hs to the moduli space M(P ) of ﬂat connections is in the class of perturbations for symplectic Floer homology considered in the proof of Theorem 3.3. In other words the restriction of CS H to the space of paths Ωφf = AΣ(Pf )/GΣ(Pf ) is the perturbed symplectic action functional and this restriction has the same critical points as CSH . They are in one-to-one correspondence with the ﬁxed points of the symplectomorphism φf,H . The class of perturbations discussed here is large enough in order to obtain nondegenerate critical points.

A Proof of Lemma 2.3

Let Q → M be a principal SO(3)-bundle over a compact 3-manifold. Through- out we denote the unit interval by I = [0, 1].

Lemma A.1 For every integer k ∈ Z there exists a gauge transformation g ∈ G(Q) with deg(g) = 2k and η(g) = 0.

Proof: The condition η(g) = 0 means that g : Q → G lifts to a map g˜ : Q → SU(2). Let ι : Σ → M be an embedding of an oriented Riemann surface. Then ι extends to an embedding of a tubular neighborhood ι : Σ × I → M. Cut out a disc D ⊂ Σ and deﬁne g˜ = 1 outside ι(D × I). Now trivialize Q over ι(D × I) and choose g˜ ◦ ι : D × I → SU(2) of degree k with g˜ ◦ ι(∂(D × I)) = 1. 2

Lemma A.2 Let ι : Σ → M be an embedding of an oriented Riemann surface ∗ with w2(ι Q) = j ∈ {0, 1}. Then there exists a gauge transformation g : Q → SO(3) of degree j whose parity

η(g) = ηΣ : π1(M) → Z2 is given by the intersection number of a loop with Σ.

33 Proof: Extend ι to an embedding of a tubular neighborhood ι : Σ × I → M. ∗ If w2(ι Q) = 0 deﬁne g ◦ ι(z, t) = g0(t) for z ∈ Σ and t ∈ I where g0(t) is a nontrivial loop in SO(3) with g0(0) = g0(1) = 1. Extend by g(p) = 1 outside ι(Σ × I). ∗ If w2(ι Q) = 1 deﬁne g(p) = 1 for p ∈/ ι(Σ×I) as above. Moreover decompose Σ as Σ = Σ1 ∪ Σ2 where Σ1 = D ⊂ Σ is a disc and Σ2 = cl(Σ \ D). Choose sections σj : Σj × I →

Q|ι(Σj ×I) for j = 1, 2 such that

2πiθ 2πiθ σ2(e ) = σ1(e )g0(θ).

Here g0(θ) is the loop in SO(3) covered by eπiθ 0 γ(θ) = ∈ SU(2). 0 e−πiθ Deﬁne g on ι(Σ × I) by

g ◦ σj (z, t) = g0(t), z ∈ Σj , t ∈ I.

−1 This is consistent with the patching condition since g0(θ)g0(t)g0(θ) = g0(t). We prove that h = g · g lifts to h˜ : Q → SU(2) and that h˜ is of degree 1. To see this note that

h˜ ◦ σj (z, t) = γ(2t), z ∈ Σj , t ∈ I, and h˜(p) = 1 for p ∈/ ι(Σ × I). Now choose a continuous function a + ib c β(r, t) = ∈ SU(2), 0 ≤ t ≤ 1 ≤ r ≤ 2. −c a − ib such that β(1, t) = γ(2t), β(r, 0) = β(r, 1) = β(2, t) = 1l, a2 + b2 + c2 = 1 and c ≥ 0. (Contract the equator over a hemisphere.) Moreover, assume that σ1 : D × I → Q extends to the disc D2 of radius 2 with the overlap condition 2πit 2πit σ2(re ) = σ1(re )g0(t) for 1 ≤ r ≤ 2. Also assume, up to homotopy, that h˜ ◦ σ2(z, t) = 1 for z ∈/ D2 and

h˜ ◦ σ2(z, t) = β(|z|, t), 1 ≤ |z| ≤ 2, t ∈ I. By the overlap condition this implies a + ib ce2πiθ h˜ ◦ σ (z, t) = γ(θ)β(r, t)γ(θ)−1 = 1 −ce−2πiθ a − ib for 1 ≤ |z| ≤ 2 and t ∈ I. Hence it follows from the deﬁnition of β that h˜ ◦ σ1(z, t) = 1 whenever |z| = 2 or t = 0, 1. Thus h˜ ◦ σ1 deﬁned a map from D2 × I/∂(D2 × I) → SU(2). Since c ≥ 0, this map is of degree 1. 2

34 Lemma A.3 If g ∈ G(Q) with deg(g) = 0 and η(g) = 0 then g is homotopic to 1.

Proof: If η(g) = 0 then g lifts to SU(2) = S3. Hence the statement follows from the Hopf degree theorem. 2 Proof of Lemma 2.3: Statement (1) follows immediately from Lemma A.3. We prove statement (2). By (H1) every cohomology class η : π1(M) → Z2 can be represented by ﬁnitely many embedded oriented Riemann surfaces ιj : Σj → M. The associated gauge transformations gj : Q → G constructed in Lemma A.2 all satisfy (1). This together with Lemma A.1 implies that for 1 every k ∈ Z and every η ∈ H (M; Z2) with k ≡ w2(Q) · η (mod 2) there exists a gauge transformation g ∈ G(Q) such that deg(g) = k and η(g) = η. Conversely, we must prove that every gauge transformation g ∈ G(Q) satis- ﬁes (1). By Lemma A.2 we may assume that η(g) = 0. This implies that g lifts to a map g˜ : Q → SU(2) and hence deg(g) is even. We prove statement (3). By (H2) there exists an embedding ι : Σ → M of an ∗ oriented Riemann surface such that w2(ι Q) = 1. So it follows from Lemma A.2 that there is a gauge transformation of degree 1. 2

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